# Gibbs

It is easy to implement this sampler:

## Julia program for Bivariate Gibbs sampler

## author: weiya <[email protected]>

## date: 2018-08-22

function bigibbs(T, rho)

x = ones(T+1)

y = ones(T+1)

for t = 1:T

x[t+1] = randn() * sqrt(1-rho^2) + rho*y[t]

y[t+1] = randn() * sqrt(1-rho^2) + rho*x[t+1]

end

return x, y

end

## example

bigibbs(100, 0.5)

Example:

We can use the following Julia program to implement this algorithm.

## Julia program for Truncated normal distribution

## author: weiya <[email protected]>

## date: 2018-08-22

# Truncated normal distribution

function rtrunormal(T, mu, sigma, mu_down)

x = ones(T)

z = ones(T+1)

# set initial value of z

z[1] = rand()

if mu < mu_down

z[1] = z[1] * exp(-0.5 * (mu - mu_down)^2 / sigma^2)

end

for t = 1:T

x[t] = rand() * (mu - mu_down + sqrt(-2*sigma^2*log(z[t]))) + mu_down

z[t+1] = rand() * exp(-(x[t] - mu)^2/(2*sigma^2))

end

return(x)

end

## example

rtrunormal(1000, 1.0, 1.0, 1.2)

In my opinion, we can illustrate this algorithm with one dimensioanl case. Suppose we want to sample from normal distribution (or uniform distribution), we can sample uniformly from the region encolsed by the coordinate axis and the density function, that is a bell shape (or a square).

Consider the normal distribution as an instance.

It is also easy to write the following Julia program.

## Julia program for Slice sampler

## author: weiya <[email protected]>

## date: 2018-08-22

function rnorm_slice(T)

x = ones(T+1)

w = ones(T+1)

for t = 1:T

w[t+1] = rand() * exp(-1.0 * x[t]^2/2)

x[t+1] = rand() * 2 * sqrt(-2*log(w[t+1])) - sqrt(-2*log(w[t+1]))

end

return x[2:end]

end

## example

rnorm_slice(100)

A special case of Completion Gibbs Sampler.

Let's illustrate the scheme with grouped counting data.

And we can obtain the following algorithm,

But it seems to be not obvious to derive the above algorithm, so I wrote some more details

Then he argues that

$m$

copies of $\mathbf y_{mis}$

in each iteration is not really necessary. And briefly summary the DA algorithm:Another example:

It seems that we do not need to derive the explicit form of

$g(x, z)$

, if we can directly obtain the conditional distribution. We can use the following Julia program to sample.## Julia program for Grouped Multinomial Data (Ex. 7.2.3)

## author: weiya <[email protected]>

## date: 2018-08-26

# call gamma function

#using SpecialFunctions

# sample from Dirichlet distributions

using Distributions

function gmulti(T, x, a, b, alpha1 = 0.5, alpha2 = 0.5, alpha3 = 0.5)

z = ones(T+1, size(x, 1)-1) # initial z satisfy `z <= x`

mu = ones(T+1)

eta = ones(T+1)

for t = 1:T

# sample from g_1(theta | y)

dir = Dirichlet([z[t, 1] + z[t, 2] + alpha1, z[t, 3] + z[t, 4] + alpha2, x[5] + alpha3])

sample = rand(dir, 1)

mu[t+1] = sample[1]

eta[t+1] = sample[2]

# sample from g_2(z | x, theta)

for i = 1:2

bi = Binomial(x[i], a[i]*mu[t+1]/(a[i]*mu[t+1]+b[i]))

z[t+1, i] = rand(bi, 1)[1]

end

for i = 3:4

bi = Binomial(x[i], a[i]*eta[t+1]/(a[i]*eta[t+1]+b[i]))

z[t+1, i] = rand(bi, 1)[1]

end

end

return mu, eta

end

# example

## data

a = [0.06, 0.14, 0.11, 0.09];

b = [0.17, 0.24, 0.19, 0.20];

x = [9, 15, 12, 7, 8];

gmulti(100, x, a, b)

Let us illuatrate this algorithm with the following example.

Last modified 4yr ago